Internal problem ID [9808]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 53.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+y \tan \relax (x )-a \left (1-a \right ) \left (\cot ^{2}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 70
dsolve(diff(y(x),x)=y(x)^2-y(x)*tan(x)+a*(1-a)*cot(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\frac {\sin \relax (x ) c_{1} a}{c_{1} \sin \relax (x )+\sin ^{2 a}\relax (x )}-\frac {\left (\sin ^{2 a}\relax (x )\right ) a}{c_{1} \sin \relax (x )+\sin ^{2 a}\relax (x )}-\frac {\sin \relax (x ) c_{1}}{c_{1} \sin \relax (x )+\sin ^{2 a}\relax (x )}\right ) \cos \relax (x )}{\sin \relax (x )} \]
✓ Solution by Mathematica
Time used: 2.298 (sec). Leaf size: 141
DSolve[y'[x]==y[x]^2-y[x]*Tan[x]+a*(1-a)*Cot[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \cot (x) \left (-1-i \sqrt {\frac {1}{a}+\frac {1}{1-a}-4} \sqrt {a-1} \sqrt {a} \left (1-\frac {2 c_1}{\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {\frac {1}{a}+\frac {1}{1-a}-4} \sqrt {a-1} \sqrt {a}}+c_1}\right )\right ) \\ y(x)\to \frac {1}{2} i \left (\sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}+i\right ) \cot (x) \\ \end{align*}